# Verification of the conditional odds ratio bound

The accepted theorem is the sharp bound ceil(2^(d+1)/3)-d-1 for the
dimension left by fixed feasible interior one-way margins and all surviving
conditional pairwise odds ratios on a binary support. It includes
uniform-margin attainment, equality supports for d>=3, and a nonvacuous
five-dimensional example for four variables. The original AIM question
about arbitrary odds-ratio specifications remains unresolved.

The written proof proceeds by classical mixed coordinates, a pivot-deletion
lemma retaining affine rank, the attributed Kostochka-Johnson-Entringer
vertex theorem, and an extra-vertex argument for the equality case.
All assumptions, including positivity on the exact support, are explicit.

Two independently written exact programs agree on 65,805 nonempty supports
in dimensions two through four. The first uses primitive integer row
elimination with safe-integer assertions. The second uses Python Fraction
Gauss-Jordan elimination and a different square generator, checks every
mask and verifies pivot deletion and unchanged margin rank. Fifty
deterministic higher-dimensional supports are additional tests, not an
exhaustive higher-dimensional computation. No repeated run is counted
as additional mathematical evidence.

At d=4 the deficiency counts 0,...,6 are 11953, 19260, 18996, 11038,
3680, 592, 16. The extremal sets have no surviving ratios, as disclosed.
The separate eleven-cell example has exactly one square, uniform
margins, ratio 768/5329 and fiber dimension five. Its integer probability
numerators sum to 2628 and each margin numerator is 1314.

The final source is checked with the desktop compiler and its publication
PDF is separately exported and visually inspected before release. Actual
compiler, artifact hashes and visual-review receipts are recorded in the
package manifest. Mathematical self-audit is not peer review or formal
verification. No proof assistant or independent human reviewer is claimed.
Known mixed-coordinate theory, the cube theorem and FPR's d=3 examples
are not presented as newly discovered. Novelty of the extremal statistical
transfer remains undetermined, and no absolute priority is claimed.